Lesson 01: Digital Systems and Number Systems

To convert any base-r number to decimal:

• Expand base-r number using the positional scheme
• Perform computation using decimal arithmetic

Base-r system to decimal system:

1. Determine the positional value of each digit (this depends on the position of the digit and the base of the number system)
2. Multiply the obtained positional values (in step 1) by the digits in the corresponding positions.
3. Sum the products calculated in Step 2. The total is the equivalent value in decimal.

Use the Radix Divide Method

The equation for decimal to base-r number:

Integral Part

1. Divide (integral part) by the base- r number:
   
2. Take the remainder as a coefficient
3. Take the quotient and repeat the division, until the quotient is equal to zero
   

Fractional Part

1. Multiply the number by the base- r number
   
2. Take the integer (either 0 or 1) as a coefficient
3. Take the resultant fraction and repeat the multiplication
   

Decimal

Characteristics of the decimal number system are as follows:

• Also called denary
• Base-10
• Uses decimal digits: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
• Positional System — position gives power to the base
• Example:
  3845.2 = (3 x 10³) + (8 x 10²) + (4 x 10¹) + (5 x 10⁰) + (2 x 10^-1)
• Positions: …5 4 3 2 1 0 . -1 -2 -3 …

The base-10 system is a Positional Number System. it means each digit has a different value according to its position. For example: 255 is (2 x 100) + (5 x 10) + (5 x 1). There are a lot of other numeral systems that do not work that way. One of the most famous is Roman numerals, which is not really a positional system. In the Roman number system, The X stands for 10, the V stands for 5, and the I is a one. Twenty-seven in Roman numerals is XXVII ; it is 10 + 10 + 5 + 1 + 1 = 27 in decimal.

Binary

All the information is stored in a computer using digital signals that translate to binary numbers. A single bit can represent two possible states: on (1) or off (0).

Characteristics of the binary number system are as follows:

• Base-2
• Digits: {0, 1}
• Binary Digits are called bits
• Positional system
• Example:
  1101.1₂ = (1 x 2³) + (1 x 2²) + (0 x 2¹) + (1 x 2⁰) + (1 x 2^-1)

Octal

Characteristics of the binary number system are as follows:

• Base-8
• Digits: {0, 1, 2, 3, 4, 5, 6, 7}
• A group of 3 binary digits can be used to represent each octal digit
• Positional system
• Example:
  357₈ = (3 x 8²) + (5 x 8¹) + (7 x 8⁰)

Hexadecimal

Characteristics of the hexadecimal number system are as follows:

• Base-16
• Digits: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F}
• Groups of 4 bits represent each base 16 digit
• Positional system
• Example:
  1F4₁₆ = (1 x 16²) + (F x 16¹) + (4 x 16⁰)

BCD

BCD (Binary Coded Decimal) is a decimal number system with a 4-bit binary number representing each decimal digit.

Do not get confused; BCD is not the same as hexadecimal. The hexadecimal also uses a 4-bit binary to represent one hex digit. Each hex digit is valid up to (F)₁₆, representing binary (1111)₂. But the one decimal digit is only up to 9 binary (1001)₂. This means that the 4-bit binary can represent 16 numbers (2⁴); in the BCD number system, the six binary codes from (1010)₂ (decimal 10) to (1111)₂ (decimal 15) are not used.

The main advantage of BCD is that it allows easy conversion between decimal and binary forms. However, the disadvantage is that the BCD code is wasteful as the states between (1010)₂ and (1111)₂ are not used.

For example, give the BCD representation for the decimal numbers 90 and 873.

The BCD representation of the number 90 is written as follows:

(90)₁₀ = (1001 0000)_BCD

For the number 873:

(873)₁₀ = (1000 0111 0011)_BCD

Points to Note:

• BCD is More Space Consuming: Each decimal digit is represented by four binary digits in BCD, which makes it less space-efficient than pure binary representation.
• BCD is Easier for Human Interpretation: BCD is closer to the decimal system and is often easier for humans to read and interpret than binary, which is why it is used in certain applications like digital clocks and calculators.
• Use in Digital Systems: BCD is frequently used in digital systems where a decimal display or decimal arithmetic is required.

Gray Code

A Gray Code encodes numbers so that adjacent numbers have one single digit differing by 1. Frank Gray invented this coding technique; thus, it is named so. The term gray code often refers to a "reflected" code, specifically the binary reflected Gray code.

Gray Code is a non-weighted code that does not ascribe a specific weight to each bit position. It is not used for arithmetic calculations.

Gray code is used in Karnaugh maps and the design of logic circuits. They also find application in rotary encoders, where the predisposition to errors increases with the number of bits that change logical states between two consecutive positions.

ASCII Code

ASCII code (American Standard Code for Information Interchange) has seven bits, representing 2⁷ = 128 symbols.

Table 1: ASCII Table

Converting a decimal number to binary in a digital logic course or general computing involves a systematic method. The process is straightforward but requires attention to detail. Here's a step-by-step guide to converting a decimal number to its binary equivalent:

Repeating Pattern in Fractional Values:

If a repeating pattern emerges, you can stop the conversion and indicate the repetition using a bar over the repeating digits. Example: Convert 0.3 to binary.

Multiplications:

• 0.3 × 2 = 0.6 (integer part 0)
• 0.6 × 2 = 1.2 (integer part 1)
• 0.2 × 2 = 0.4 (integer part 0)
• 0.4 × 2 = 0.8 (integer part 0)
• 0.8 × 2 = 1.6 (integer part 1)
• 0.6 × 2 = 1.2 (integer part 1)
• ... (pattern repeats 0011)
• Binary equivalent: 0.0100110011... ≈ (0.010011)

Some binary fractions may not terminate or have a repeating pattern. In those cases, you'll need to decide on an appropriate level of precision for your application.

To combine the integer and fractional parts, place a binary point between them. Practice regularly to become proficient in these conversions.

Converting numbers from binary to decimal in a digital logic course involves understanding the base-2 system and how it translates to the base-10 system we commonly use. Here's a step-by-step guide with examples:

Binary Number System:

• Base-2: Binary is a base-2 numeral system. Each digit in a binary number is called a bit and can only be a 0 or a 1.
• Positional Value: Each position in a binary number has a value that is a power of 2, starting from 0 on the right (integer) and -1 on the right (fraction).

Conversion Steps: Binary to Decimal:

1. List Down the Powers of 2: Write down the powers of 2 starting from right to left for integer, beginning with 2⁰, and from right to left after the decimal point starting with 2^-1.
2. Multiply Each Binary Digit with Its Power of 2: For each digit in the binary number, multiply it by the corresponding power of 2.
3. Sum the Results: Add all the products together. The sum is the decimal equivalent of the binary number.

Convert (1101.01)₂ to decimal.

(1101.01)₂ = (1 x 2³) + (1 x 2²) + (0 x 2¹) + (1 x 2⁰) + (0 x 2^-1) + (1 x 2^-2) = (13.25)₁₀

• 8 = 2³
• Each group of 3 digits in binary represents an Octal digit

• 16 = 2⁴
• Each group of 4 binary bits represents a Hexadecimal digit

Convert to Binary as an intermediate step

Example: (26.2)₈ to Hex

Works both ways (Octal to Hex & Hex to Octal)

Binary-to-BCD Conversion

To convert binary to BCD, we will first convert the binary number to decimal and then the decimal number to BCD number. Similarly, to convert BCD to binary, we will first convert the BCD to decimal and then convert the decimal to equivalent binary.

Step-by-Step Guide for Binary to BCD Conversion:

1. Convert the Binary Number to Decimal
   • First, convert the binary number to its decimal equivalent by multiplying each bit by the corresponding power of 2 and summing them up.
   • For example, convert binary 10110 1100 to decimal.
     Binary 1101100 is 1×26 + 1×25 + 0×24 + 1×23 + 1×22 + 0×21 + 0×20 = 64 + 32 + 0 + 8 + 4 + 0 + 0 = 108 in decimal.
2. Convert the Decimal Number to BCD
   • Write down each decimal digit as a separate 4-bit binary number (since each decimal digit ranges from 0 to 9, which can be represented by 4 binary digits).
   • Separate 108 into 1, 0, and 8.
   • Convert to binary:
     1 is 0001 in binary.
     0 is 0000 in binary.
     8 is 1000 in binary.
   • BCD representation: (0001 0000 1000)_BCD.

Binary-to-Gray Code Conversion

Gray code is a binary numeral system where adjacent numbers differ by only one bit. It is used in error correction, position encoding, and communication systems to minimize errors due to bit changes.

Binary to Gray Code Conversion

The conversion process from binary code to Gray code involves the following steps:

1. MSB (most significant bit) or the leftmost bit of the given binary is the same as MSB of the Gray number.
2. To get the next bit of Gray code, perform XOR (exclusive OX) MSB with the next bit of binary number (move from left to right). Write the sum and discard the carry.
3. Repeat step 2 for all the bits in the sequence till LSB.

Gray to Binary Code Conversion

The steps given below are required to be followed to convert a given Gray value into its binary equivalent:

1. MSB does not change. So, write the leftmost bit of Gray code as the MSB of binary code.
2. Now, perform XOR recently achieved binary digit with the next adjacent Gray code bit. The sum must be written as the next bit of binary equivalent, while the carry must be discarded.
3. The above-discussed step must be followed for all the bits present in the sequence.

The 1-bit XOR can be represented as adder logic.

   
Figure 2: Gray code Conversion using adder

Ex1: Octal to Decimal

Ex2: Hexadecimal to Decimal

Ex3: Base-5 to Decimal

Ex4: (315.312510)₁₀ to Octal

Sign-Magnitude

The most significant bit (MSB) is used as the sign bit in the sign-magnitude approach. The rest of the bits represent the magnitude of the number.

• Positive Numbers:
  To represent a positive number, you write its binary form and make sure the MSB is 0.
  Example: To represent +5 in an 8-bit sign-magnitude form:
  • Decimal 5 in binary is 101.
  • To make it 8 bits, add zeros at the beginning: 00000101.
  • The MSB is 0, indicating a positive number, so +5 is 00000101.
• Negative Numbers:
  To represent a negative number, you write the binary form of its absolute value and ensure the MSB is 1.
  Steps to Represent a Number in Sign-Magnitude:
  • Convert the absolute value of the number to binary.
  • Set the MSB to 0 for positive numbers or to 1 for negative numbers.
  Example: To represent -5 in an 8-bit sign-magnitude form:
  • Decimal 5 in binary is 101.
  • To make it 8 bits, add zeros at the beginning: 00000101.
  • Change the MSB to 1 to indicate a negative number, so -5 is 10000101.

Key Points:

• Bit Number: The number of bits determines the range of representable values. For example, with 8 bits, you can represent numbers from -127 to +127.
• Zero Representation: Sign-magnitude allows for two representations of zero: +0 (0000 00000) and -0 (1000 0000), which is considered a drawback of this system.
• Arithmetic Operations: Arithmetic operations (addition, subtraction, etc.) using sign-magnitude can be more complex than with other representations like two's complement.

One's Complement

In one's complement, negative numbers are represented by inverting all the bits of their positive counterparts.

• Positive Numbers: Represented normally with the MSB as 0.
  • Positive numbers in one's complement are represented just like in regular binary notation.
  • Example: The number +5 in an 8-bit binary system is represented as 00000101.
• Negative Numbers: To represent a negative number in one's complement:
  • First, write the positive version of the number in binary.
  • Then, invert all the bits (change 0s to 1s and 1s to 0s).
Example: To represent -5 in an 8-bit one's complement form:
• Write +5 in binary: 00000101.
• Invert all bits: 11111010.
So, -5 in one's complement (8-bit) is 11111010.

Key Points:

• Zero Representation: One's complement, like sign-magnitude, also allows for two representations of zero: +0 (0000) and -0 (1111).
• One's complement can lead to complications in arithmetic operations due to these two representations of zero.
• Range of Representable Values: With n bits, you can represent numbers from -(2^(n-1)-1) to +(2^(n-1)-1).
• Arithmetic Operations: Arithmetic operations using one's complement can be simpler than sign-magnitude but still have some complexities.

Two's Complement

Two's complement is a system for representing negative integers in binary. It simplifies arithmetic operations and avoids the issues of having two representations of zero, which occurs in one's complement.

• Concept: Represents negative numbers by inverting the bits of the corresponding positive number and adding 1.
• Positive Numbers: Represented normally with the MSB as 0.
• Negative Numbers: Invert all bits of the positive number and add 1.
  • Steps to Represent a Negative Number:
    To represent a negative number in two's complement:
    1. Start with the binary representation of the positive number.
    2. Invert all bits (change 0s to 1s and 1s to 0s).
    3. Add 1 to the inverted result.
    Example: To represent -5 in an 8-bit two's complement form:
    
    1. Write +5 in binary: 00000101.
    2. Invert all bits: 11111010.
    3. Add 1: 11111010 + 1 = 11111011.
    So, -5 in two's complement (8-bit) is 11111011.

Faster Method (Shortcut):

• Start from the Right: Scan the binary number from the right (LSB).
• Copy Zeros: Copy all zeros to the 2's complement result.
• Find First 1: Stop copying when you encounter the first 1.
• Invert Remaining Bits: Invert all remaining bits (including the first 1) to get the 2's complement.

Integer Part

Fractional Part

Additional Notes:

• To convert a 2's complement value back to its negative binary form, follow the same steps in reverse.
• Practice with different negative binary numbers to solidify your understanding.
• Visualize the bit inversion and addition process to grasp the concept better.

Binary addition follows the same principles as decimal addition but with only two digits (0 and 1).

Rules of Binary Addition:

Example:

Binary multiplication is simpler than decimal multiplication because each digit is either 0 or 1.

Rules of Binary Multiplication:  There are four rules of binary multiplication.

Example:

The binary division is similar to decimal division but simpler due to its binary nature. It is called the long division procedure.

Example:

Example: Convert the following floating-point values to IEEE 754 Single-Precision Format